Multiscale Variance Stabilization
Variance stabilization is one of the oldest techniques in statistics, see, for example, Barlett (1936). Suppose one has a random variable X, which has a distribution dependent on some parameter θ and that the variance of X is given by var(X) = σ2(θ). If X was Gaussian with distribution X ~ N(μ, σ2) and the parameter of interest is μ, then it is patently clear that the variance σ2 does not depend on μ. For many other random variables, this is not true. For example, if Y was distributed as a Poisson random variable, Y ~ Pois(λ), then the (only) parameter of interest is λ and the variance σ2(λ) = λ. In other words, the variance depends directly on the parameter of interest, λ, which is also the mean. Hence, we often refer to a ‘mean-variance’ relationship, and for Poisson variables the variance is equal to the mean.
KeywordsHaar Wavelet Piecewise Constant Variance Stabilization Poisson Random Variable Wavelet Shrinkage
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